Mesh-Independent Operator Preconditioning for Boundary Elements on Open Curves

Abstract

Boundary value problems for the Poisson equation in the exterior of an open bounded Lipschitz curve $\mathcal{C}$ can be recast as first-kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators (BIOs). Based on the recent discovery in [C. Jerez-Hanckes and J. Nédélec, SIAM J. Math. Anal., 44 (2012), pp. 2666--2694] of inverses of these BIOs for $\mathcal{C}=[-1,1]$, we pursue operator preconditioning of the linear systems of equations arising from Galerkin--Petrov discretization by means of zeroth- and first-order boundary elements. The preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes.